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# 醫用流體力學 - PowerPoint PPT Presentation

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### 醫用流體力學

Arterial Fluid Dynamics

Physiological Fluid Dynamics

Arterial Pressure

Away from the heart

• Conduct blood flow from Left ventricle (LV) to peripheral organs

• Aortic valve  Aortic arch (180° turn)

• Geometry changes :Tapering

• Geometry changes : Branching

• Mechanical properties changes

Stress-Strain relations of agerabbit’s thoracic aorta

• Mass Conservation

• Conservation of momentum

• Conservation of energy

Background age

• Fundamental VariablesPressure、 Flow

• Geometrical VariablesSize、 Thickness 、 Length、 Curvature

• Mechanical Properties Stiffness 、Visco-Elasticity

• Consider a conduct filled with incompressible fluid of density  and pressure p, let u be the only non-zero velocity component

Poiseuille age’s Law (1840)

• Assume steady flow, u= u(r ) with no body forces, the equation of motion

• Rate of flow through the tube

• Mean velocity of flow

• Shear stress at the wall

• Skin friction

• Shear stress in terms of skin friction

Implication of Poiseuille age’s Law

• Q is proportional to the fourth power of the radius.

• Q is directly proportional to the pressure difference.

• Q is inversely proportional to the length of the tube.

• If the arteries becomes constricted, the bloodpressure requires to supply the blood flow adequately will risesubstantially,leading to the state of hypertension.

Optimum design of Blood Vessel Bifurcation (Poiseuille age’s formula)

For a given pressure drop, 1% change in vessel radius results in a 4% changes in flow

Murray (1926)

Rosen (1967)

Work done

Metabolism Energy loss

The optimum vessel radius is proportional flow to the 1/3 power, and

Minimize P at the bifurcation point B

An optimum location B would be

for arbitrary movements of B.

The optimum is obtained when

C-B direction

The optimum is obtained when

Similarly, displaced B along D-B direction, we find

The continuity equation gives

We find

which is often referred to as Murray’s Law

37.5°

Let a ageo denotes the radius of the aorta, and assume equal bifurcation in all generation

If the capillary blood vessel has a radius of 5 um and the radius of the aorta is 1.5 cm.

We find n=30.

The total number of blood vessel is about

230109.

Note: in fact arteries rarely bifurcation symmetrically (a1=a2).

For human, only one symmetric bifurcation.

For dog, there are none.

• Consider pulsatile flow in a circular vessel, p=p(x, t) and u = u(r, t)

• For a sinusoidal flow

• The general solution of the ODE in the form involves Bessel functions of complex arguments

U(r=a)=0 (non-slip)

U(r=0)=finite

• Introducing Womersley number 

• As 0, the velocity profile becomes parabolic.

• As , viscosity is negligible U(r)=-i P/.

• From Poiseuille’s Law

the flux is proportional to the pressure difference (p1-p2). However, the blood flow in veins are remarkably non-linear.

• The flow in elastic conduct gradually attains a maximum value as the pressure difference increases and then on longer increases.

• Axial velocity, v

• Lumen area, S

• Let T denotes the tension of the blood vessel per unit thickness, wall thickness h, vessel radius a

• Let ro be the radius of zero tension state, the Hooke’s Law gives elastic constant E as

• Consider steady flow in elastic tube of length L, assume the tube is long and the pressure is function of axial coordinate z, let P1 and P2 denote the inlet and outlet pressure and the external pressure surrounding the tube is P0

• Assume the flow through the tube obey Poiseuille’s law, the flow becomes

• Consider inviscid and incompressible fluid flow in elastic tube of lumen area A,

• By linearizing the equations

• Combining the continuity and momentum equations,

• The wave equations

Pulse Wave Velocity (PWV)

• Constitutive relationship between aortic volume and pressure where K is the volume elasticity of the aorta, and V0 is the end-systolic volume.

• If the aorta is very soft (K is very small), let I(t) and Q(t) denote the inflow and outflow rates, we have

• During diastole, the aortic valve is closed and there is no flow into the aorta. Hence I(t) =0. where is a non-invasive measure aortic volume elasticity. Let Td be the duration of diastolic phase, the aortic pressure (Pd) at the end of this phase or just prior to ejection is given by

• The volume elasticity that depicts the exponential drop of aortic pressure is given by

Reynolds #; Strouhal #; Womersley flow into the aorta. Hence I(t) =0. where

• Reynolds number

• Strouhal number

• Womersley number

Flows under the action of Oscillating pressure gradient flow into the aorta. Hence I(t) =0. where

Wave propagation in Blood Vessel flow into the aorta. Hence I(t) =0. where

• Pulse wave propagation in arteries

• A(x, t) depends on transmural pressure,

Here c is the wave propagation velocity.

For thin walled elastic tube: flow into the aorta. Hence I(t) =0. where

• Consider the elasticity of the tube, arterial diameter  blood pressure

For a thick walled elastic tube:

Balance of Force in Arterial Wall flow into the aorta. Hence I(t) =0. where

Resonant vibration of flow in a circular tube flow into the aorta. Hence I(t) =0. where

• When the tube length is equal to the half wave length

• This is called the fundamental frequency of the natural vibration.

• Hemodynamics : Effects of Frequency on the Pressure-flow relationship of Arterial tree

Boundary conditions flow into the aorta. Hence I(t) =0. where

Pressure-Flow flow into the aorta. Hence I(t) =0. where

Mean Velocity Profile flow into the aorta. Hence I(t) =0. where

(Dog Aorta)

Velocity waveform at the upper descending aorta of a dog flow into the aorta. Hence I(t) =0. where

Effect of Womersly number on the velocity distribution flow into the aorta. Hence I(t) =0. where

Blood Pressure flow into the aorta. Hence I(t) =0. where Evolution

Effect of sinusoidal pressure wave speed of various frequencies on the instantaneous aortic pressure

Distribution of Atherosclerotic Sites in Human frequencies on the instantaneous aortic pressure

Stress Concentration frequencies on the instantaneous aortic pressure

Conditions

Atherosclerotic disease at the carotid bifurcation frequencies on the instantaneous aortic pressure

Stress contours in arterial branching

What frequencies on the instantaneous aortic pressure’s the blood flows in reality?

• Non-uniform geometry

• Bifurcations

• Non-Newtonian

• Viscoelastic wall

• Fluid-solid interactions

In vitro measurement of artery pressures and flows frequencies on the instantaneous aortic pressure

Ultrasonic flowmeters frequencies on the instantaneous aortic pressure

### Electromagnetic flowmeters frequencies on the instantaneous aortic pressure

Electromagnetic flowmeters have existed for measurement of blood flow rate outside the body during open heart surgery.

This miniature probe is for acute and chronic, low flow measurements in small animals and rodents.

Sizes from 1 to 10 mm internal circumference

Electromagnetic flowmeters frequencies on the instantaneous aortic pressure

Electromagnetic flowmeters frequencies on the instantaneous aortic pressure

• Faraday's principle of electromagnetic induction can be applied to any electrical conductor (including blood) which moves through a magnetic field. The electromagnetic blood flowmeter is sometimes used during vascular surgery to measure the quantity of blood passing through a vessel or graft, before during or after surgery. A circular probe with a gap to fit the vessel is fitted around the vessel. This probe applies an alternating magnetic field across the vessel and detects the voltage induced by the flow via small electrodes in contact with the vessel.

Electromagnetic flowmeters frequencies on the instantaneous aortic pressure

• Alternating magnetic fields (typically at 400 Hz) are used since the induced voltages are in the microvolt region and d.c. electrode potentials may cause significant errors with unchanging magnetic fields. A number of probes are required to fit the various diameters of blood vessel.

• An alternative design carries the sensing device on the tip of a special catheter which passes inside the vessel and generates a magnetic field in the space around it and has the electrodes on its surface.

SQUARE-WAVE ELECTROMAGNETIC frequencies on the instantaneous aortic pressureBLOOD FLOWMETERS

Pulse Oximetry frequencies on the instantaneous aortic pressure

• Takuo Aoyagi(1974) developed the principle of pulse oximetry. The next year, Nihon Kohden introduced the world's first ear oximeter, OLV-5100, which used pulse oximetry to noninvasively measure saturated blood oxygen without the need to sample blood. All pulse oximeters today are based on Dr. Aoyagi's original principle of pulse oximetry.

CCA Root frequencies on the instantaneous aortic pressure

### Variation of velocity waveforms across the arterial vessel frequencies on the instantaneous aortic pressure

Common Carotid Artery (CCA)

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

CCA frequencies on the instantaneous aortic pressure

Radial Artery frequencies on the instantaneous aortic pressure

0.23 cm in diameter

Brachial Artery frequencies on the instantaneous aortic pressure

Brachial Vein frequencies on the instantaneous aortic pressure